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Many of the arguments regarding the early Double at -2,-2 begin with the statement "assume that your opponent will never lose his market..."
and end with "...thus we have shown that you should double at just the point where you could lose your market." My interpretation is "if you assume that the correct strategy is never to lose your market then you reach the conclusion the the correct strategy is never to lose your market." In my early days as a wanna-be mathematician, assuming something to be true to then prove it to be true is strictly verboten. For the *true* mathematicians among you (and even you fellow wished-I'd-beens, am I right or wrong here? -- Chuck Bower Keep in mind that you are not winning 2 while your opponent can only win 1. That your opponent would win the game with a centered cube means that your opponent would roll a market loser at some point without doubling. Even though you may be playing on for the gammon, you should expect your losses to be doubled to 2. A priori, there is no reason you can't correctly lose your market and play on for the gammon, reaching a position that is No Double/Pass. What must happen is that your checker play is the same as for DMP, and that you never cash: Either you regain your market, or you win an undoubled gammon. Your mwc must equal your gwc at DMP. However, I don't believe you can be perfectly confident that you will never have to cash (say, after your last checker is hit and contained, which has positive probability from any closeout), or that your opponent will not be able to run off the gammon, which happens with positive probability from any anchor. Therefore, I don't believe that you can ever lose your market without a theoretical error, making your match winning chances lower than your game winning chances. If you play someone whose checker play is as for DMP, who never loses his/her market, and who always takes, then you can't win more than 50%. You will lose 50% of the games, and every time you lose the cube will be on 2. If some of your wins are only 1 point, you are worse off. If you ever make a checker play that is not the same as for DMP, then you are worse off. The strategy I mentioned is semi-perfect, winning at least 50% against anyone. It is not perfectly exploitive. Although there are clear exploitive improvements here (such as passing in 100% lost positions), no set strategy is perfectly exploitive. There might be some opponent who goes crazy if you misplay an opening 3-1, but that is not a good argument for misplaying. However, if you use a strategy that wins less than 50% against this semi-perfect one, your strategy is not theoretically correct. It is not new to suggest delaying the double at 2-away 2-away. Many strong players do this in practice, and I won a minimatch in Las Vegas because I delayed the double, and my opponent won a single game. This tactic is mentioned in Kit's article on the 5-point match, and in the following articles in the rec.games.backgammon archive. Ron Karr, Walter Trice, Albert Steg, Darse Billings, Chuck Bower, Robert Koca -- Douglas Zare Here is a quick proof that, in theory, doubling at your first opportunity is at least as good as any other strategy: If you double at your first opportunity, your opponent will have a take, and your match-winning chances will be your % chances of winning the game. If you don't double at your first opportunity, you present your opponent with an option. He may: a) Double on the next roll. If he does, your match-winning chances are again equal to your % chances of winning the game. It didn't matter whether or not you doubled. b) Not double on the next roll. He will (in theory) adopt this option only if it is at least as favorable to him as doubling. Thus, if he chooses not to double, his match-winning chances after not doubling must be >= his game-winning chances -- otherwise, he would have doubled. So if he (correctly) doesn't double, your match-winning chances will be <= your game-winning chances. Therefore, if you do not double, your match-winning chances will never be > your game-winning chances, while if you do double they are exactly = your game winning chances. So, doubling always results in a match-winning percentage which is >= the match-winning percentage from not doubling. The above analysis is just a theoretical one, of course, since it does assume that your opponent is playing properly. If that assumption is not accepted, then of course it may be more profitable to not double immediately. -- Kit Woolsey Is there a way to rigorously prove that every game will reach a position which meets both of these conditions? If not, what is the proper strategy if the first position you reach with a market loser also contains a sequence leaving you worse than -1:-2? -- Matt Reklaitis Yes, there is a fairly simple proof using mathematical induction. It goes as follows: Let's say that both players are playing perfectly, which means that they will double as soon as they have a market loser. Rollouts have shown that after turn 1 there are no market-losing sequences. Let's suppose that on turn N a player has a market-losing sequence, but he also has a roll with would cause him to have a pass of his opponent's cube. If this is the case, then by definition his opponent had a market-losing sequence on turn N-1, so would have doubled. Therefore, the hypothetical situation you describe cannot occur. I should add that the above proof does not take into account the possibility of gammons. Gammons complicate the issue. Several years ago Chris Yep showed me a very long and rigorous proof that even taking gammons into account it is impossible to reach a position where a player has a market-losing sequence but it is incorrect for him to double. I'm afraid I didn't keep a copy of the proof (or if I did, I don't know where it is). Maybe you can talk Chris into reproducing this proof if you are interested. -- Kit Woolsey I agree with your proof, but it only proves 1 of the conditions: that a market losing sequence must have previously existed. But how do you simultaneously prove the other condition: that opps market loser occured without there also existing a sequence leaving him worse than -2:-1. -- Matt Reklaitis Kit showed that at the first market loser, there can't be a roll that leads to Double/Pass. That would mean there was an earlier market loser, which would contradict the assumption that we were looking at the first market loser. With correct play, there is no first undoubled market loser, so there are no undoubled market losing sequences. To include gammons, consider when the first time is that the mwc is different from the gwc at DMP. -- Douglas Zare That's where mathematical induction comes in. Remember, I said initially that rollouts have shown that on the first response there are no possible market-losing sequences. Let's suppose that the Nth turn is the first turn of the game where there is a possible market-losing sequence for player A. If player A has a horror roll which would make him worst 2 to 1 underdog (i.e. give him a pass), then on player B's previous turn player B had a market-losing sequence -- namely whatever he rolled along with player A's horror roll. But we assumed that this Nth turn is the first time there is a market-losing sequence on the horizon. Therefore player B did not have a market-losing sequence on the previous roll, so player A can't have a horror roll which makes him worse than 2 to 1. -- Kit Woolsey I agree...but a roll isn't a sequence. Couldn't it be possible that at the first market loser, while I can still take opp's cube after all my rolls, that there are still sequences (I roll + opp rolls) that leave me worse than -2:-1. I dont see how the induction handles this. Perhaps I am misunderstanding the -2:-1 condition. -- Matt Reklaitis That is even easier. If you roll one of those bad rolls (where you still have a take, but after your opponent's next roll you could be worse than at -1 -2, what do you think your opponent will then do? Obviously he will turn the cube, since he has the market losers. So you can never dodge the bullet, therefore it is correct for you to double with market losers even if your bad rolls may give you the potential to be worse than -1 -2 if your opponent rolls well. -- Kit Woolsey Thread 2 (Cube Strategies at different skill levels) I sort of go by gut feel based on how strong I perceive my opponent to be, how complicated the position is to play for his/her side, and how much of the match is at stake if I take (factoring in gammon chances). My very basic approach to something like this would have a first pass as follows: 1) If I have difficulty deciding whether this position is a technical take or not and it doesn't seem more difficult than "normal" to play from my somewhat weaker opp's side, I simply pass. 2) However, I didn't think this position was borderline, and honestly was nearly spot on estimating it to be a .8 take. While I didn't like the over 30% of the times I estimated I get G'd and nearly put out of the match. 3) Assuming I was playing a clearly weaker player. If it is really easy to play for his side, I'd honestly pass it. For example, if the position is easy to play when the side on roll gets a succession of good shots and my side dances, I would pass it. However, it gets more complicated if I enter. Therefore, it's a pass for me vs an intermediate who can play the position well, but perhaps a take against one who cannot. But there's the recube to consider. When my opp's takepoint is lower than normal and I have less recube vig than normal, I think I'd pass vs most average intermediates. These are some of the things one needs to think about when playing a match and both players need to be aware of any real skill differences. -- Neil Kazaross My general guidelines (being on the intermediate side) are don't double when far ahead, double in volatile gammonish positions. I used to double and take aggressively on races, but now I try to play them straight up because my knowledge of racing equities has improved. If I play at an expert level on races, should I still be aggressive to make up for the skill disparity on the complicated games? One thing I've noticed. Against top players, I tend to have lopsided wins and losses and other players are neck and neck. Maybe that's just a mental illusion. -- Tad Bright In qualitative terms, try to put more match equity at stake when the position is less complicated than "normal," or when the skill difference is more favorable for you. That you are now comfortable with racing positions is an argument for being more aggressive with cube actions. Whether a position is more complicated than normal or not depends on the cube level. Oddly, this is because the meaning of normal changes. You should be more willing to be aggressive with the cube (doubling early and taking late) when the cube is high, if you don't feel that you have a skill advantage. That's because by encouraging the escalation of the cube, you are skipping more of the match where you would be at a disadvantage, disproportionately more than the additional equity you may be giving up because the cube is higher. In some sense, the net errors you skip grow as the square of the match equity at stake, while the error you make now is proportionate to the match equity at stake. There are scores at which being aggressive tends to lead to more complicated remainders of the match. You may want to avoid taking aggressively if that puts more match equity at stake immediately, but tends to make the rest of the match more complicated. An example is 2-away 3-away. As the leader, if you take an initial double, you will usually end up trailing Crawford 2-away, which is pretty tough to play for both sides. I think DMP tends to be easier for an advanced player to handle against an expert; advanced players do not win enough gammons trailing Crawford 2-away. So, against a stronger player, you may wish to be more conservative with your takes leading 2-away 3-away, since you may not be much of an underdog at 2-away 2-away, which you should convert to DMP. -- Douglas Zare 1) In races the weaker player should cube more aggressively since there's almost no skill involved. 2) In an easy to play and gammonish blitz position, the weaker player should also cube more aggressively, and then if in doubt as to the proper checker play, just bang away. However, the weaker player must realize that if the stronger opp. survives the blitz, the play can become very complicated and he may end up outplayed. 3) In a complicated yet gammonish position like a long backgame defence, the weaker player shouldn't, IMHO, get overly aggressive with the cube since he's likely to produce a whopper or two in the checker play department and will be outplayed by his opp. 4) However, in an easy to play position such as bearing off against a well timed backgame, the weaker player should cube aggressively since the game is very easy to play when almost all of your moves are forced and just a matter of dice as to whether he gets hit or not. If most of the weaker player's wins are gammons and he's in doubt as to whether to cube the well timed backgame (assuming he's bearing off or almost bearing off) he should simply just cube it. -- Neil Kazaross "Question on H-W match equity table" Your basic takepoint on a 2-cube, almost regardless of the score, is usually right around the normal money 25% mark. There are some exceptions like -3 -3 or trailing -4 -3 where you need more to take, but 25% is a pretty good rule of thumb. Last weekend in Charlotte, I found another score where this 25% rule doesn't work. I was trailing 7-1 to 11 (or -4 -10 if you prefer), the opponent had offered me the cube and I was trying to decide whether to take or pass. When you do the match equity math at this score, you'll find that the trailer's basic takepoint is 33%! Drop and you're down 1-8 with 13% ME, take and lose 1-9 with 7% ME, take and win makes it 3-7 which is 25% ME. Is this because of rounding or is there something about this score that makes the trailer's takepoint so high? -- John O'Hagan Being 3-away, relatively speaking is not a great spot, due to the fact that four point games (cube on 4 or gammons on a 2-cube) waste 25%. So trailer doesn't mind (so much) leader getting to the 3-away score. Having said that, the recube vig needs to included before you reach your final conclusion. At this score, the recube vig might make up a lot of this deficit. Even though leader can use all four points, he can't use 8 (gammon on a 4-cube) but trailer can. What I've noticed is that when the score if fairly close (I think leads of 1-3), the recube vig doesn't make up for the raw deficit but when the difference is 4 or more it really kicks in. -- Chuck Bower To determine the importance of rounding, look at what happens when you adjust the entries by 0.5%. After all, an entry of 13% could mean anything from 12.5% to 13.5%. The most sensitive entry will be the entry for passing. Rather than risking 6% to gain 12%, perhaps you are risking 5.5% to gain 12.5%. That still comes out to a dead-cube take point of about 30%. It is hard for the leader to take advantage of the doubling cube at 3-away many-away. I think that argues for being less willing to take initial doubles trailing 4-away many-away. It is very important for the trailer to have a good recube at some match scores. It tends to be less important for the leader to have good opportunities to recube. -- Douglas Zare |